Question

A steady two-dimensional flow field is specified by the stream function \[ \psi = k x^3 y, \] where \( x \) and \( y \) are in meter and the constant \( k = 1 \, \text{m}^2 \text{s}^{-1} \). The magnitude of acceleration at a point \( (x, y) = (1 \, \text{m}, 1 \, \text{m}) \) is ________________ m/s² (round off to 2 decimal places).

Hint:

In steady two-dimensional flow, the acceleration can be calculated using the velocity components obtained from the stream function and the corresponding partial derivatives.

Answer 1

Correct Answer: 4.2

The magnitude of the acceleration in a steady flow field is given by the formula: \[ a = \sqrt{ \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial v}{\partial y} \right)^2 + 2 \left( \frac{\partial u}{\partial y} \right)^2 + 2 \left( \frac{\partial v}{\partial x} \right)^2 } \] where:
- \( u \) is the velocity component in the \( x \)-direction,
- \( v \) is the velocity component in the \( y \)-direction.
The velocity components can be obtained from the stream function using the following relations: \[ u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}. \] Given the stream function: \[ \psi = k x^3 y, \] where \( k = 1 \, \text{m}^2 \text{s}^{-1} \), we compute the velocity components: \[ u = \frac{\partial \psi}{\partial y} = k x^3 = x^3, \] \[ v = -\frac{\partial \psi}{\partial x} = -k \cdot 3 x^2 y = -3 x^2 y. \] At the point \( (x, y) = (1, 1) \), we have: \[ u = 1^3 = 1 \, \text{m/s}, \quad v = -3(1)^2(1) = -3 \, \text{m/s}. \] Next, we compute the acceleration components. First, calculate the partial derivatives: \[ \frac{\partial u}{\partial x} = 3 x^2, \quad \frac{\partial v}{\partial y} = -3 x^2. \] At \( (x, y) = (1, 1) \), we have: \[ \frac{\partial u}{\partial x} = 3(1)^2 = 3, \quad \frac{\partial v}{\partial y} = -3(1)^2 = -3. \] Now, calculate the acceleration: \[ a = \sqrt{ (3)^2 + (-3)^2 } = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \, \text{m/s}^2. \] Thus, the magnitude of the acceleration at the point \( (x, y) = (1, 1) \) is \( \boxed{4.24} \, \text{m/s}^2 \).